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Evaluate the following logarithm

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2 Answers

Properties of Logarithms:
logex = ln x 
ln e = 1
 
a) logx(ab)= logx(a)+ logx(b)
b) logx(a/b)= logx(a) - logx(b)
c) logx(an) = n * logx (a)
 
so... log2e3 = (ln 2) + (3 * ln e) (properties a and c)
        2loge3= ln (32) = ln 9 (reverse of property c)
        
       ln 2 + 3*ln e + ln 9 = 3* ln (9*2*e) (combine the two and use property a )
       
       3*ln 18e - ln 18 =3 ln (18e/18) = 3 ln e  (combine with 3rd eq, use property b and simplify)
      since ln e = 1, 3 ln e = 3
 
 final answer = 3
ln(2e^3)+2ln(3)-ln(18)
ln(2)+ln(e^3)+2ln(3)-ln(18)
0.693147+3+2*(1.098612)-2.890371 (from tables)
0.693147+3+2.197224-2.890371
3.693147+2.197224-2.890371
5.890371-2.890371
3
or ln(2)+ln(e^3)+ln(3)+ln(3)-ln(18)
    ln(2)+ln(3)+ln(3)-ln(18)+ln(e^3)
    ln(2*3*3)-ln(18)+ln(e^3)
    ln(18)-ln(18)+ln(e^3)
    ln(18/18)+ln(e^3)
    ln(1)+ln(e^3)
    0+ln(e^3)
    ln(e^3)
    3